Science of science has become a popular topic that attracts great attentions from the research community. The development of data analytics technologies and the readily available scholarly data enable the exploration of data-driven prediction, which plays a pivotal role in finding the trend of scientific impact. In this paper, we analyse methods and applications in data-driven prediction in the science of science, and discuss their significance. First, we introduce the background and review the current state of the science of science. Second, we review data-driven prediction based on paper citation count, and investigate research issues in this area. Then, we discuss methods to predict scholar impact, and we analyse different approaches to promote the scholarly collaboration in the collaboration network. This paper also discusses open issues and existing challenges, and suggests potential research directions.